Concept Partition Pattern
http:// ontologydesignpatterns.org/wiki/Submission:Partition
Olaf Noppens
Inst. of Artificial Intelligence Ulm University
Germany
1 Introduction
ThePartition Patternis a logical pattern that introduces axioms which model a partition of concepts. A partition is a general structure which is divided into sev- eral disjoint parts. With respect to ontologies the structure is a concept which is divided into several pair-wise disjoint concepts. This pattern reflects the simplest case where a named concept is defined as a partition of concepts.
2 Pattern
2.1 Problem
ThePartition Pattern describes how to model a partition, i. e., a named con- cept which is divided into several disjoint concepts. Applying this pattern to an ontology will introduce the necessary axioms.
2.2 Solution
Let P be a named concept that is the partition which is divided into several conceptsCi. Then the partition is defined by introducing the following axioms (expressed in the standard DL syntax) as also be shown in Figure 1:
P ≡C0C1 · · · Cn andCiCj=⊥for 0≤i, j≤n, i=j.
Note that some ontology languages such as OWL 2 [1] provides disjointness ax- ioms as syntactical sugar to make the definition of pair-wise disjointness easier.
In OWL 2 the pattern can be translated into the following axioms (expressed in OWL 2 abstract syntax):
EquivalentClasses(P ObjectUnionOf(C0, C1, ..., Cn)) DisjointClasses(C0, C1, ..., Cn)
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Fig. 1.Graphical representation of thePartition Pattern. Here,EquivalentClassesand DisjointClasses are axioms and ObjectUnionOf is a class constructor building a dis- junction over an arbitrary number of conceptsC0, ...,Cn.
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2.3 Example
Consider a world where only humans and animals live. Then the inhabitants of this world are partitioned into humans and animals. The following two axioms EquivalentClasses(Inhabitant ObjectUnionOf(Human, Animal))
DisjointClasses(Human, Animal)
describe the partitions of inhabitants into human and animals.
3 Pattern Usage
In an ontology about family relationship we defined concepts such asPerson, AuntandParentOfSonwhich are characterized by a relationships such ashasChild (resp. the inverse relationshiphasParent),hasSibling,married-withas well as by the gender of people (Malerespectively Female). There are a lot of sim- ilar ontologies about family relationships. Our version can be downloaded at http://www.informatik.uni-ulm.de/ki/Noppens/generation.owl.
EquivalentClasses(Parent-Of-Son
ObjectSomeValuesFrom (has-Child Male) ) EquivalentClasses(Parent-Of-Daughter
ObjectSomeValuesFrom(has-Child Female) )
EquivalentClasses(Aunt ObjectIntersectionOf(Uncle-Or-Aunt Female)) EquivalentClasses(Uncle-Or-Aunt
ObjectIntersectionOf(Person
ObjectSomeValuesFrom(has-Sibling Parent)))
The conceptGenderis partitioned in Maleand Female. Applying this pattern results in the following axioms:
EquivalentClasses(Gender, ObjectUnionOf(Male Female)) DisjointClasses(Male Female)
4 Summary and Future Work
ThePartition Pattern describes how to model a partition. The pattern reflects the simplest case where anamedconcept is the partition of (arbitrary) concepts.
Future work will be concerned with a more general variant of this pattern: in some cases, the partition concept is not explicitly named (i. e., is not a named concept) but implicitly used, for instance, as value range of quantifications. In other words, no equivalent class axiom will be used but the value range is the union of the pair-wise disjoint parts of the partition.
References
1. Motik, B., Patel-Schneider, P.F., Parsia, B.: OWL 2 Structural Specification and Functional-Style Syntax. W3C Candidate Recommendation 11 June 2009 (June 2009)
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